If xy + z = x(y + z), which of the following must be true?

BTGmoderatorDC wrote:If xy + z = x(y + z), which of the following must be true?

A. x=0 and y=0
B. x=1 and y=1
C. y=1 and z=0
D. x=1 or y=0
E. x=1 or z=0

OA E

Source: GMAT Prep

The key word here is must.

Take, for example, consider answer choice A.
While it's possible that x = 0 and z = 0, it need not be the case.
For example, x=1, y=1 and z=1 is a solution to the equation. So, this eliminates A.

My Approach:
Given: xy+z = x(y+z)
Expand: xy+z = xy + xz
Subtract xy from both sides: z = xz
Rearrange: xz - z = 0
Factor: z(x-1) = 0

This tells us that z = 0 OR x = 1
Answer: E

Cheers,
Brent

BTGmoderatorDC wrote:If xy + z = x(y + z), which of the following must be true?

A. x=0 and y=0
B. x=1 and y=1
C. y=1 and z=0
D. x=1 or y=0
E. x=1 or z=0

OA E

Source: GMAT Prep

Simplifying, we have:

xy + z = xy + xz

z = xz

xz - z = 0

z(x - 1) = 0

z = 0 or x = 1

Answer: E